An examination of the methods shows, at best, the claim should probably be worded a little more carefully.
"In recent decades, rich black kids have been more likely to go to prison than poor white kids."
The measure of wealth
The measure of wealth in the study was total assets minus total debt for individuals in the cohort in 1985. At that time, the individuals in the cohort were 20-28 years old. This measure did include assets co-owned with a spouse, but, importantly did not include parental assets. "Rich kids" vs. "poor kids" probably doesn't accurately communicate the variable of interest here. For example, when I hear "rich kid", I'm more likely to think of parental wealth than individual wealth. Additionally, I would expect assets minus debt at 20-28 to be inversely correlated with education in many cases. Not going to college and working a full time low wage job, for example, would often give you more wealth during this age range than going to college, and education is associated with incarceration rate.
The measure of incarceration
Incarceration was measured indirectly. Interviews were conducted annually from 1979-1994 and biennially afterwards. The primary method for determining incarceration history was whether the respondent was incarcerated during the interview or interview attempt. There were direct questions in the 1980 survey, but responses to these questions would have only contributed to identifying prior incarceration (which disqualified a respondent from this analysis). Recall, this analysis looked at the relationship between individual wealth in 1985 and future incarceration.
Sample vs. population
The claim in both the Washington Post report of the article is ostensibly about the overall US population. The NLSY79 cohort is stated to be a nationally representative cross section. This is not supported with any data or a reference. This particular analysis uses a subgroup -- individuals who were not previously incarcerated. That seems like a reasonable choice, given the association between prior incarceration and both future incarceration and future wealth, and the goal of investigating the relationship between pre-incarceration wealth with future incarceration. However, because prior incarceration at 20-28 is also strongly associated with race, it is a choice that will bias the sample and prevent generalization to the overall US population.
Additionally, and probably most importantly, no inferential statistics were reported. The only measures reported were descriptive statistics of the sample. There is no confidence interval
A reasonable conclusion:
The only scientifically valid conclusion one can draw from this particular study is one about the sample itself. To make a claim about the overall US population using a representative sample, inferential statistics should be used. Any sample statistic comparing incarceration rates of groups based on race and wealth should include a confidence interval in order to draw conclusions about the population. If inferential statistics were applied to these data, the conclusion should be carefully worded to reflect the actual variables being measured, rather than simply rich, poor, and go to prison (vs. be in prison at any of several follow up interviews).
Importantly, though, despite not providing sufficient evidence for a reasonable conclusion, these data do suggest a possible racial component to incarceration that acts beyond an individual's wealth in young adulthood. It means the question should be evaluated more robustly, or, perhaps, that the analysis isn't finished. It may not be worthwhile to complete the analysis with these exact data, given the problems with how incarceration was defined.
A note on normalizing incarceration data by "crimes committed"
A highly upvoted comment suggested that the study should have "look(ed) at incarceration normalized by crimes committed". The truth of who has committed a crime and who hasn't, is, perhaps unfortunately, not an available statistic. It is generally not good practice to normalize for something you cannot accurately and precisely measure. It would be very nice if we could just know who committed a crime and who didn't, but in that world the criminal justice system would be very different indeed, and there would not be much need for this study or this question at all.